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Last night, in the deep darkness that comes at this time of year, I got in the car and drove to an empty parking lot outside of a shopping mall. I circled around until I found the row of large metal donation bins in the corner farthest from the mall entrance; parked; opened the trunk of the car, lugged three heavy bags of children’s books to one of the bins, and tossed the books by handfuls into the clanging interior.

I estimate that I’ve read some of those books hundreds of times, talking to my young sons about the pictures, savouring the rhymes and alliteration, laughing at slapstick humour and hoping for the baby animals to be reunited with their mothers. (If you haven’t read children’s books lately, that’s a staggeringly common theme .)

We’ve set aside the books we’ll be shipping to Boston in a few weeks. Some are treasured gifts from family and friends; others are simply too delightful and beloved to part with.

(Kinda skipped over that move to Boston, didn’t I? I’ll get to that in a minute.)

When you’re moving across both a continent and a border, every single item gets evaluated. Is it worthy of shipping? Of packing into storage? Either way, you’ll be paying for the privilege of holding on to your possessions. So there’s a new calculus that gets invoked, beyond that wonderful KonMari question – does it spark joy? – and beyond its monetary value.

It doesn’t always matter how much you paid for it, and how much joy it sparks; if it won’t fit in the shipping container and it’s going to cost you an extra fifty bucks a month to store it, you will confront the question of how much it’s worth to you right now.

(This isn’t another ode to living minimally, by the way. It’s simply a letter from this moment in my life to this moment in yours.)

So about this move to Cambridge: my partner David has been offered a totally amazing job. He’s going to be teaching at Harvard. That’s particularly amazing because David isn’t an academic – so we didn’t see this coming. It was definitely not in the plans. But it is a what-what-omg-can’t-say-no opportunity. So we are moving, with our two young kids, to a new city.

And we’re working every day to focus on the adventure of it all, and the possibilities it presents, and not get swept away by the grief of saying a thousand goodbyes, of giving away our sons’ baby clothes, of leaving our beloved home.

I’m trying not to be swallowed by sorrow when the books our children teethed on clang into an echoing metal bin in a dark corner of an empty parking lot.

Not because I won’t allow myself to be sad. I will. But because the aching beauty of this experience is that it is bringing me face to face with the passage of time.

Face to face with the present. (This is what transition looks like. This is how not-knowing feels. This is the best use for this object right in this specific moment, no matter what it meant to me last week or six months ago. This is the sound of books falling into a bin, no matter how much my mind wants to fashion it into a leitmotiv of nostalgic farewells.)

Face to face with the transitory value of all material things. (The boys have outgrown the baby clothes. Those books will be better loved elsewhere. And still, the grief is here.)

And for that, I am thankful.

This is a letter from this moment in my life to this moment in yours. What does this moment hold for you?

What are you holding on to because it matters enough to keep investing in it?

What are you holding on to that has served its purpose and is ready to be released?

What do you need to grieve in the cold dark? What memories will keep you warm forever?

Where is the not-knowing that makes you so uncomfortable that you reach for your soothing balm of choice again and again?

A few months back, my mathematician friend Greg Martin at the University of British Columbia invited my feedback on a paper he was writing on how to increase the number of women speaking at math conferences. One bit in particular jumped out at me:

He used statistical probability to disprove the notion that underrepresentation of women on any given speaker’s list “just happens.”

If you’ve ever followed a debate about why an event has so few women speakers, you’re likely familiar with the argument that gender was not a factor (AKA “we chose the best speakers, regardless of gender”), and that speakers were chosen in an unbiased fashion, on merit alone. Well, if I understand the math correctly, the odds of that assertion being true are next to nothing.

What follows is, to be sure, a fairly technical read—but it’s an accessible and engaging one, too. I hope you’ll follow along, even if math isn’t your most cherished subject, and share this with your favourite stats nerds.

Over to Greg:

In a recent article, I made the following statement regarding the genders of plenary speakers at the International Congress of Mathematicians:

The appropriate null hypothesis is “the ICM speakers were selected independently of gender from among the pool of people who have received PhDs in mathematics in the last 25 years”. Under our conservative 24% assumption from above, the observation of nineteen male plenary speakers and one female plenary speaker rejects ( ) this null hypothesis. Indeed, it is 18 times as likely that we would have seen an “overrepresentation” of female plenary speakers (five or more, since ) by chance than to have seen at most one.

Clearly the gist of this statement is that having one female speaker out of twenty is really unlikely to be the result of chance or bad luck. But how were those exact numbers, 0.031 and 18, determined? How can we calculate analogous numbers in similar situations? It turns out not to be that hard, once we know the formula to use; the purpose of this post is to supply that formula and give some examples of how to use it.

Let’s start by examining an idealized situation divorced from social issues. Imagine that we have a giant bag full of marbles; the marbles come in two colours, orange and green (we like the orange ones better), which are well mixed together. We are going to take 50 marbles out of the bag, one by one, and see how many orange marbles we end up with.

Of course, without knowing whether orange marbles are common, rare, or somewhere in between, we have no idea how many orange marbles to expect! Let’s say that we know that 40% of the marbles are orange and 60% of them are green. On average, we’d expect to get orange marbles in our selection of 50 marbles; but of course, we might get lucky and get more than 20 orange marbles, or we might get unlucky and get fewer than 20 marbles. How likely is it to end up with, say, only 13 orange marbles?

As it happens, the probability of ending up with exactly 13 orange marbles, if we draw 50 marbles from a bag containing 40% orange marbles and 60% green marbles, is1

Here, the symbol is a “binomial coefficient”2 (sometimes written as , and pronounced “50 choose 13”).

While it’s not impossible to calculate this on our own, we might as well use WolframAlpha to help out: if we type in

Binomial[50,13] * (0.4)^13 * (1-0.4)^(50-13) ,

we receive the answer , telling us that the chance of getting exactly 13 orange marbles is about 1.47%. We can also go to Stat Trek’s online binomial calculator and enter 0.4, 50, and 13 in the first three fields; we see the answer 0.0147378… appear in the fourth field.

The general version of the above situation is: we have independent opportunities for success or failure (in the example above, was 50, and “success” meant drawing an orange marble while “failure” meant drawing a green marble). In each opportunity, the probability of success is some number (above, ). If we are interested in a certain number, , of successes (above, was 13), then the probability of succeeding exactly out of times is given by the formula3

Now, suppose we suspected that some funny business was going on. For example, maybe our housemate loves orange marbles, and we think that she snuck around one night and pulled a bunch of orange marbles out of the bag before we started taking our 50 marbles. (I guess the bag is so huge that emptying it out and counting all the marbles it holds is out of the question.) She denies it, however, saying “sometimes you get few orange marbles by random chance”. What should you believe?

It seems unlikely to get only 13 orange marbles out of 50 (if there really are still 40% orange marbles in the bag). On the other hand, any specific number of orange marbles is pretty unlikely. Getting exactly 20 orange marbles is the most likely outcome, as we remarked above, but even that has less than an 11.5% chance of happening4. So instead of asking how likely it is to get exactly 13 orange marbles, standard procedure is to ask how likely it is to get at most 13 orange marbles (in other words, how likely it is to get a result this extreme or even more extreme).

There’s no secret here: to get the probability of obtaining at most 13 orange marbles out of 50 (assuming that the bag really does contain 40% orange marbles), we just add up the probability of obtaining 0 orange marbles, 1 orange marble, 2 orange marbles, and so on up to 13 orange marbles:5

Fortunately WolframAlpha can do this for us, if we enter6

the sum of Binomial[50,j] * (0.4)^j * (1-0.4)^(50-j) from 0 to 13 .

The Stat Trek binomial calculator also calculates this for us (the answer appears in the third field from the bottom). Either way, we obtain 0.0279883…; so the chances of getting at most 13 orange marbles is less than 2.8%.

So… what should we believe about our suspicious housemate? Well, there are no guarantees in any of this: while 2.8% is a pretty small probability of getting at most 13 orange marbles out of 50 if everything is on the up and up, it can still happen—in fact, it happens about 1 time in 36. But that low a probability seems unlikely. If this were a statistics paper, our “null hypothesis” would be that nothing funny had happened to the bag of marbles, and we would probably reject that null hypothesis since the standard threshold is 5% (the ubiquitous ” level” of statistical tests). If the only other reasonable hypothesis, in your mind, was that your housemate was stealing orange marbles during the night, then perhaps you should believe that.

More often, though, we get to make similar observations many different times. (I’m not sure what’s going on in this world we’re inventing, but maybe we get a well-proportioned marble delivery every week to top up the bag of marbles, and we draw a 50-marble allotment every week…?) If we consistently see extreme results like this, then we become convinced that something fishy is happening. So each individual calculation becomes a piece of data that we can collect to see if there is a larger, systemic pattern.

From a sufficiently abstract perspective, examining the gender of speakers at an STEM conference is the same as looking at how many marbles of each color we get from the bag. The collection of all people who might have been invited to this conference is our bag of marbles; female speakers are our orange marbles (our “successes”) and male speakers are our green marbles (our “failures”—only in the context of examining appropriate representation of women in STEM, that is!). In a world where gender was unrelated to being invited to STEM conferences, we would expect the proportion of female speakers to be (more or less, since things are always a little bit random) the same as the proportion of female practitioners in the field of the conference.

Of course, knowing or estimating that latter proportion can be difficult. In the quote that started this post, I used the estimate 24% (which I called a conservative estimate) for the proportion of women in research mathematics; my reasoning was based on data of PhD graduates in the US over the past 25 years, where women earned at least 24% of the mathematics PhDs in all 25 of those years (and sometimes up to 34%).7

The particular conference I was discussing had 20 plenary speakers, only one of whom was female. The above formulas tell us that the probability of having at most one female speaker by chance is

,

or a tiny bit over 3%. On the other hand, if gender bias were not present in the academic system, then we would expect 20-speaker conferences to have, on average, female speakers. The probability of having women “overrepresented”—that is, of there being at least 5 female speakers—is

or over 54%. Since , it is indeed almost 18 times more likely (under our assumptions) to have an “overrepresentation” of female speakers than to have at most one.

Although it doesn’t show all the numbers corresponding to its calculations, the Conference Diversity Distribution Calculator gives a nice visual representation of how many female speakers we should expect at conferences in a bias-free world.

Footnotes

Coders and other detail-oriented folks might raise the following nitpick: if the first marble we draw is orange, say, then the percentage of orange marbles remaining in the bag is then actually slightly less than 40%! While certainly true, the slight change is negligible if our huge bag contains a very large number of marbles. It is common practice to ignore this detail when “selecting without replacement”, as a statistician would say, from a huge number of possibilities.

For number geeks and other interested rockstars: in general, the binomial coefficient is a shorthand for a quotient of factorials: Here the factorial is just the product of all the numbers up to , that is, See http://en.wikipedia.org/wiki/Combination for more information.

into WolframAlpha, or the numbers 0.4, 50, and 20 into the Stat Trek binomial calculator.

In general, the probability of getting at most successes out of tries, if each success occurs with probability , is
which we can write using summation notation (or Sigma notation) as follows: The probability of getting at least successes out of tries equals

Or, for lovers of syntax,

Sum[ Binomial[50,j] * (0.4)^j * (1-0.4)^(50-j), {j,0,13} ] .

As a side note: among all critical responses to the posting of that article, I found that this one detail—the 24% figure—generated the most criticism, much of it (in my opinion) demonstrably misguided. Some people complained that the “correct” figure to use should be much lower, perhaps around 10-12%, because that is the proportion of women among tenured faculty at top-tier mathematics departments. There are at least two flaws in this argument. First, it assumes that being hired and granted tenure at top-tier institutions is done equitably for women and men, but there is ample evidence of systemic gender bias in hiring, one that is more pronounced at top-tier institutions; in other words, it relies upon the assumption that men at top-tier institutions are generally stronger than women at next-tier institutions, which a dubious assumption. Second, my article explicitly discusses all of the obstacles for women in STEM, even after they earn their PhD—implicit bias in evaluations, double standards for behaviour and self-promotion, the impostor phenomenon, and all the usual factors known to contribute to the leaky pipeline—and hence comparing the proportion of female PhD-earners to the proportion of female conference speakers is reasonable. In this context, claiming that the lower proportion of women in top-tier tenured positions should be the measuring stick for representation at conferences is, quite simply, using a symptom of the very problem we are discussing to justify perpetuating the problem.

But I do know what to do when she shows up. And that’s made all the difference.

How to tell if your Comparer is a problem

Here’s the thing: I used to make the Comparer’s observations all about me, in a negative way. I’d turn “She’s courageous and creative” into “…and I’m not.” “He’s an inspired teacher” became “…and I could use more training.”

Now, there’s nothing inherently wrong with feeling like I’d benefit from more teacher training. The problem lies in the tone and the impact of this inner script.

The tone of my Comparer is pretty harsh. And the impact of her words is… deflating.

Depleted, lacking, and less worthy are pretty awful places to work from. Imagine trying to set a new fitness target from that emotional place. Or dreaming up a new creative project. Or setting some new career goals.

Would you feel motivated? Passionate? Committed?

Yeah. Didn’t think so.

The two best medicines for comparison

In the words of my wise friend Alexis Morgan, “Comparison disconnects us from what we’re actually passionate about and makes it hard to make decisions that deeply fuel our fire.”

Here’s how I observe the mental process: First, I notice a quality in someone that I admire. (Comparison can go in the other direction, too, but that’s a topic for another post.) Courage, artistry, boldness, and athleticism are common ones for me; yours will likely be different.

Next, I do an on-the-spot assessment of how I measure up in that capacity. Luckily (ha), I have intimate knowledge of all the times in my life I have failed to live courageously, creatively, boldly, or athletically, so I quickly come up with a long list of demerits.

Cue depletion, lack, and unworthiness.

But wait! Help is on the way, in the form of two key tools: Grounding and Inquiry.

Grounding is where we begin.

The first thing we need to do, when we’re stuck in depletion, lack, and unworthiness, is to become present and grounded in what’s real. To do that, we have to drop the stories flying around in our monkey minds, reconnect with our bodies and our breath, and remember who we really are.

We spend a good deal of time on this in Beyond Compare. (For those of you who are new around here, that’s the self-study program I created with the fabulous Tanya Geisler.) Grounding is where we begin; it’s where we close; and it’s the foundation on which the entire program is built. The simple but essential kindness of being with ourselves in the present moment is the first step towards feeling whole – and dropping the comparison trip.

(Isn’t “trip” the best hippie word ever? We must bring it back.)

Inquiring without judgment

And then, once we’ve found our roots and remembered ourselves, we can inquire into what lies beneath the comparison. When I envy someone’s athleticism, it’s often because I’m feeling like caring for my body comes last on my priority list. (Raise your hands, mothers of infants and toddlers.) The cure for that doesn’t need to be a 50km trail run; it could be making myself a healthy breakfast rather than toast-on-the-go, again. Or taking the baby out for a walk while I listen to one of my favouritepodcasts. Or showering with some gorgeously scented soap.

When I’m gobsmacked by someone’s creativity, I’m typically not making time for my own creative pursuits.

When I’m awed by their courage, I’m probably taking the path of least resistance somewhere in my life.

You get my drift.

The inquiry process is at the heart of Beyond Compare, and obviously I can’t cover the whole thing in a single blog post, but the essence of it is to notice the quality that you’re admiring, and then find the places within yourself where you are and aren’t expressing that quality – and, importantly, note them without judging yourself.

When you approach comparison this way, your Comparer will stop seeing the fabulous people in your world as objects of envy, jealousy, or other kinds of win/lose dynamics – and start seeing them as founts of inspiration. And I don’t just mean inspiration in the sense that Jen’s crazy trail runs are INSPIRING (though of course they are), but the deeper kind of inspiration where you actually feel inspired to live your life according to your own fiery passions – not those of other people.

Do you want to strip naked, cover yourself in body paint, and recreate a Damien Hirst statue with your own pregnant body? Then great – g’head and take copious notes from Ms. Palmer’s fantastic art performance. But if what you really want is to express your own creativity, boldness, and courage, then don’t bother trying to be like her. Ground yourself, and inquire within. (Deep within.)

A new tool to help you quiet your Comparer

Because – to paraphrase one of my very favourite sayings – what the world needs isn’t more people who’ve done living-statue performance art, or endurance trail runs. What the world needs is people who have come alive.

If the Comparer is tripping you up in depletion, lack, and unworthiness, Beyond Compare can help you reconnect with your own unique fire and move forward with grounded focus.

I’m so proud of this program and of the positive impact it’s had for people. If you recognize your own Comparer in any of what I’ve written, I would love for you to take part in the newest addition to our body of work on comparison:

It’s designed to take just 5 minutes a day, for 5 days. And it will give you a real taste of what life beyond compare looks like for you. I don’t know what that will be – but I do know it’s uniquely yours. And more of your uniqueness is exactly what the world needs.

Hi. I'm Lauren.

I'm a seasoned tech entrepreneur and author who asks a lot of questions. I offer strategy and coaching for creatives, entrepreneurs, and accidental bosses who want to infuse more joy, curiosity, and ease into their lives and work – and create growth that has meaning and purpose.
More about me…

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